Source: p 77. Sweet Reason: A Field Guide to Modern Logic (2010 2 ed) by Henle, Garfield, Tymoczko.
I pursue only intuition; please do not answer with formal proofs or Truth Tables. I already comprehend, and so ask not about, the direct proof. Instead, why is this true intuitively? The citation below does not convince me intuitively, because it does not explain why intuitively, ¬(P ↔ Q) is not instead: ¬P ↔ Q or ¬P ↔ ¬Q.
The negation of a biconditional,
Harriet will go if and only if Gloria goes. P ↔ Q
may be less familiar, but it still makes intuitive sense. It's another biconditional,
Harriet will go if and only if Gloria doesn't go. P ↔ ¬Q
After all, P ↔ Q says P and Q have the same truth value. P ↔ ¬Q says that P and ¬Q have the same truth value, that is, that P and Q have different truth values.